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326                      ANALYTICAL MECHANICS
eliminated by taking the origin as the position of zero potential energy. Thus we have
E7=/32g2+/33<Z3+                     (XXVII')
But since the vibrations are supposed to be small, q remains a small quantity during the motion. Therefore the higher powers of q are negligible compared with q2. Thus neglecting the higher terms we obtain the following expression foi the potential energy of the system.
[7=|/3g2,                        (XXVIII)
where i j8 = ft-
The kinetic energy, on the other hand, takes the form
T=%aq\                          (XXIX)
where a is a constant and q = -5* .   But since the system is
Cut
conservative the sum of its dynamical energy remains constant. Therefore
E= T+U = i *g* + I ft*.                      (XXX)
Differentiating both sides of the last equation with respect
to the time,
aq + pq= 0,                       (XXXI)
which is the differential equation of simple harmonic motion. Therefore we have
q = a sin \    (t + fc)                 (XXXII)
!La
and                       p=27r\/~~.                           (XXXIII)
Hence the main part of Lagrange's method consists of selecting the coordinate which defines the position of the system in such a way as to make the expressions for the kinetic and potential energies of the forms